An $$\mathfrak {sl}_2$$-type tensor category for the Virasoro algebra at central charge 25 and applications

McRae, Robert; Yang, Jian

doi:10.1007/s00209-022-03197-z

Cited by 2 publications

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“…Specifically,  c is the category of C 1 -cofinite gradingrestricted generalized V c -modules, or equivalently the category of finite-length Vir-modules at central charge c whose composition factors are simple quotients of reducible Verma modules. Writing c = 13 − 6t − 6t −1 for some Î  t , the braided tensor category  c is by now well understood for Ï  t [33,34], Î   t 1 [21,[35][36][37], and t = -1 [38,39].…”

Section: Introductionmentioning

confidence: 99%

Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at (p, q)-central charge

McRae,

Sopin

2024

Phys. Scr.

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Let O_{c} be the category of finite-length modules for the Virasoro Lie algebra at central charge c whose composition factors are irreducible quotients of reducible Verma modules. For any complex c, this category admits the vertex algebraic braided tensor category structure of Huang–Lepowsky–Zhang. Here, we begin the detailed study of O_{c_{p, q}} where c_{p, q} = 1 − 6(p−q)^2/pq for relatively prime integers p, q ≥ 2; in conformal field theory, O_{c_{p, q}} corresponds to a logarithmic extension of the central charge c_{p, q} Virasoro minimal model. We particularly focus on the Virasoro Kac modules K_{r, s}, where r, s ≥1 are integers, in O_{c_{p, q}} defined by Morin-Duchesne– Rasmussen–Ridout, which are finitely-generated submodules of Feigin–Fuchs modules for the Virasoro algebra. We prove that K_{r, s} is rigid and self-dual when 1 ≤ r ≤ p and 1 ≤ s ≤ q, but that not all K_{r, s} are rigid when r > p or s > q. That is, O_{c_{p, q}} is not a rigid tensor category. We also show that all Kac modules and all simple modules in O_{c_{p, q}} are homomorphic images of repeated tensor products of K_{1, 2} and K_{2, 1}, and we determine completely how K_{1, 2} and K_{2, 1} tensor with Kac modules and simple modules in O_{c_{p, q}}. In the process, we prove some fusion rule conjectures of Morin-Duchesne–Rasmussen–Ridout.

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